3.3 Lattice Models - Minimized Free Energy

The minimized free energy model concentrates on the double well
structure of distorted perovskite structures. A one-dimensional
lattice model was developed by Omura et al [OAI91], based on
previous work on isomorphous ferroelectric phase transition by
Ishibashi, which was published in 1992 [Ish92]. It relies on
the minimization of the free energy which is modeled by a fourth order
approach for the energy of the center ion stemming from the lattice
cell and a coupling term describing the interaction between the ions
of neighboring cells:

(3.5)

is the dipole moment of the th dipole, is the
coupling coefficient. is a
function of the temperature, namely

(3.6)

is the Curie Temperature. Consequently

(3.7)

is true for the ferroelectric phase of the material, and finally it is assumed that

(3.8)

The overall polarization is given by

(3.9)

The model assumes the existence of permanent dipoles with random
distribution (Fig. 3.14). These dipoles serve as centers for
the nucleation of domains.

The two allowed dipole moments are restricted by the
model to for the positive nuclei and for
the negative ones.

The simulation starts at an equilibrium state where all the 'free'
dipoles are negatively polarized and the time-dependent expansion of
the domains that occurs around the nucleation centers is
examined. Therefore a viscosity coefficient is introduced
which takes into account the switching delay of the individual dipoles. The
resulting evolution of polarization is given by the Landau
Khalatnikov kinetic equation [Bau99]

and numerical solution of the resulting set of
(3.11). The typical shape of the resulting hysteresis is plotted in Fig. 3.15.

Figure 3.15:
Hysteresis of the free energy model

This approach was extended to two-dimensional lattice structures by
Omura et al. [OAI92] in 1992. The functional for the free energy
had to be modified in order to include the increased number of
possible coupling partners as follows

(3.13)

(3.14)

The Landau Khalatnikov equation reads then

(3.15)

This first approach did not consider the influence of the
depolarization field. Instead the electric field was assumed to be
constant in the entire simulation area. The influence of
depolarization was finally added by Baudry in 1999 [Bau99], who
implemented Poisson's equation into the system.

The basic intent of the free energy method is to gain insight into
the material properties and the main focus is the correct
qualitative reproduction of physical effects. Typical simulation
setups analyze areas of a size of about dipoles.